As \(G\)  is abelian the singularity \(\mathbb {A}^3/G\)  and its crepant resolutions are toric. There is a method of marking the exceptional subvarieties of \(G\) -Hilb – the edges and vertices in the triangulation – by characters of \(G\)  known as “Reid's recipe”. This was used to explicitly describe the McKay correspondence in the classical terms of providing a basis of \(H^*\big (G\text{\textnormal {-Hilb}}\,\mathbb {A}^3,\mathbb {Z}\big )\)  indexed by characters by Craw [1]}. It was later categorified by Logvinenko [2]}, Cautis–Logvinenko [3]}, and Cautis–Craw–Logvinenko [4]}, who expressed the locus labelled by a character \(\chi \)  in terms of the support of an object associated to \(\chi \)  in the derived category of \(G\) -Hilb. We will discuss this in more detail in Sections REF  and REF .
